Keywords: Energy conservation, Onsager conjecture, turbulence, Besov spaces. Abstract: In this talk we will discuss a possibility of constructing solutions to the forced stationary Euler equations with limit regularity. The problem is motivated by finding vector fields that enjoy the properties of a turbulent flow, i.e. anomalous energy dissipation, smooth forcing, and regularity 1/3 in a certain Besov norm. The time-dependent version of this problem is known as the Onsager conjecture. We will exhibit a number of conditions which rule out existence of such solutions. Those include, for instance, conditions on the singularity set. An example of a field with smoothness 1/3, but integrability 18/11, which is Onsager-supercritical will be presented.